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Theorem swopolem 5600
Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypothesis
Ref Expression
swopolem.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
Assertion
Ref Expression
swopolem ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑌,𝑧   𝑧,𝑍
Allowed substitution hints:   𝑌(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem swopolem
StepHypRef Expression
1 swopolem.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
21ralrimivvva 3200 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
3 breq1 5151 . . . 4 (𝑥 = 𝑋 → (𝑥𝑅𝑦𝑋𝑅𝑦))
4 breq1 5151 . . . . 5 (𝑥 = 𝑋 → (𝑥𝑅𝑧𝑋𝑅𝑧))
54orbi1d 915 . . . 4 (𝑥 = 𝑋 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧𝑧𝑅𝑦)))
63, 5imbi12d 344 . . 3 (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧𝑧𝑅𝑦))))
7 breq2 5152 . . . 4 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
8 breq2 5152 . . . . 5 (𝑦 = 𝑌 → (𝑧𝑅𝑦𝑧𝑅𝑌))
98orbi2d 914 . . . 4 (𝑦 = 𝑌 → ((𝑋𝑅𝑧𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧𝑧𝑅𝑌)))
107, 9imbi12d 344 . . 3 (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧𝑧𝑅𝑌))))
11 breq2 5152 . . . . 5 (𝑧 = 𝑍 → (𝑋𝑅𝑧𝑋𝑅𝑍))
12 breq1 5151 . . . . 5 (𝑧 = 𝑍 → (𝑧𝑅𝑌𝑍𝑅𝑌))
1311, 12orbi12d 917 . . . 4 (𝑧 = 𝑍 → ((𝑋𝑅𝑧𝑧𝑅𝑌) ↔ (𝑋𝑅𝑍𝑍𝑅𝑌)))
1413imbi2d 340 . . 3 (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧𝑧𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌))))
156, 10, 14rspc3v 3625 . 2 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌))))
162, 15mpan9 506 1 ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wral 3058   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149
This theorem is referenced by:  swoer  8755  swoord1  8756  swoord2  8757
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